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This book builds theoretical statistics from the first principles of probability theory. Starting from the basics of probability, the authors develop the theory of statistical inference using techniques, definitions, and concepts that are statistical and are natural extensions and consequences of previous concepts. Intended for first-year graduate students, this book can be used for students majoring in statistics who have a solid mathematics background. It can also be used in a way that stresses the more practical uses of statistical theory, being more concerned with understanding basic statistical concepts and deriving reasonable statistical procedures for a variety of situations, and less concerned with formal optimality investigations.
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Stalisticallnference Second fdition George Casella Roger l. Berger DUXBURY ADVANC ED SERIE S 01KONOMlKO nANEniITHMIO AeHNON BIBAloeHKH ':fQ�� Ap. 5)q.:5 Ta�. qs tao. Statistical Inference Second Edition George Casella University of Florida Roger L. Berger North Carolina State University DUXBURY • THOMSON LEARNING Australia • Canada • Mexico • Singapore • Spain • United Kingdom • United States DUXBURY t.('·;ti{:)·""';1 \:jl:' ; ! ! • � to "� � t:. ¢ �� ... � t n-IOMSON LEARNING Sponsoring Edi1;f �� numbers, that is, S = (0,00). We can classify sample spaces into two types according to the number of elements they contain. Sample spaces can be either countable or uncountable; if the elements of a sample space can be put into 1-1 correspondence with a subset of the integers, the sample space is countable. Of course, if the sample space contains only a finite number of elements, it is countable. Thus, the coin-toss and SAT score sample spaces are both countable (in fact, finite), whereas the reaction time sample space is uncountable, since the positive real numbers cannot be put into 1-1 correspondence with the integers. If, however, we measured reaction time to the nearest second, then the sample space would be (in seconds) S = {O, 1,2,3, ... }, which is then countable. This distinction between countable and uncountable sample spaces is important only in that it dictates the way in which probabilities can be assigned. For the most part, this causes no problems, although the mathematical treatment of the situations is different. On a philosophical level, it might be argued that there can only be count able sample spaces, since measurements cannot be made with infinite accuracy. (A sample space consisting of, say, all ten-digit numbers is a countable sample space.) While in practice this is true, probabilistic and statistical methods associated with uncountable sample spaces are, in general, less cumbersome than those for countable sample spaces, and provide a close approximation to the true (countable) situation . Once the sample space has been defined, we are in a position to consider collections of possible outcomes of an experiment. x ... • Definition 1.1.2 An event is any collection of possible outcomes of an experiment, that is, any subset of S (including S itself). Let A be an event, a subset of S. We say the event A occurs if the outcome of the experiment is in the set A. When speaking of probabilities, we generally speak of the probability of an event, rather than a set. But we may use the terms interchangeably. We first need to define formally the following two relationships, which allow us to order and equate sets: A c B xEA => xE B; A = B A c B and B c A. (co